Import the dataset and other packages

#rm(list = ls())

#packages
library(bayesrules)
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library(randomForest)
randomForest 4.7-1.1
Type rfNews() to see new features/changes/bug fixes.
library(rpart)
library(tree)
library(pROC)
Type 'citation("pROC")' for a citation.

Attaching package: ‘pROC’

The following objects are masked from ‘package:stats’:

    cov, smooth, var
library(mgcv)
Loading required package: nlme
This is mgcv 1.8-41. For overview type 'help("mgcv-package")'.
library(ISLR)
library(dplyr)

Attaching package: ‘dplyr’

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    collapse

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    combine

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library(tidyverse)
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── Attaching packages ──────────────────────────────────────────────────────── tidyverse 1.3.2 ──✔ ggplot2 3.3.6     ✔ purrr   0.3.4
✔ tibble  3.1.8     ✔ stringr 1.4.1
✔ tidyr   1.2.1     ✔ forcats 0.5.2
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library(faraway)

Attaching package: ‘faraway’

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library(olsrr)
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library(caret)
Loading required package: lattice

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Attaching package: ‘caret’

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#Data Set
data1 <- mtcars
summary(data1)
      mpg             cyl             disp             hp             drat             wt       
 Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0   Min.   :2.760   Min.   :1.513  
 1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5   1st Qu.:3.080   1st Qu.:2.581  
 Median :19.20   Median :6.000   Median :196.3   Median :123.0   Median :3.695   Median :3.325  
 Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7   Mean   :3.597   Mean   :3.217  
 3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0   3rd Qu.:3.920   3rd Qu.:3.610  
 Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0   Max.   :4.930   Max.   :5.424  
      qsec             vs               am              gear            carb      
 Min.   :14.50   Min.   :0.0000   Min.   :0.0000   Min.   :3.000   Min.   :1.000  
 1st Qu.:16.89   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
 Median :17.71   Median :0.0000   Median :0.0000   Median :4.000   Median :2.000  
 Mean   :17.85   Mean   :0.4375   Mean   :0.4062   Mean   :3.688   Mean   :2.812  
 3rd Qu.:18.90   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
 Max.   :22.90   Max.   :1.0000   Max.   :1.0000   Max.   :5.000   Max.   :8.000  
head(data1)



1. OLS

a) Visualize the data using Pairs to see if there is any collinearity

pairs(data1)


# Conclusion
# Some of the predictors are linearly depedent based on the graph. (ex. mpg~drat, hp~wt)

cor(data1)
            mpg        cyl       disp         hp        drat         wt        qsec         vs
mpg   1.0000000 -0.8521620 -0.8475514 -0.7761684  0.68117191 -0.8676594  0.41868403  0.6640389
cyl  -0.8521620  1.0000000  0.9020329  0.8324475 -0.69993811  0.7824958 -0.59124207 -0.8108118
disp -0.8475514  0.9020329  1.0000000  0.7909486 -0.71021393  0.8879799 -0.43369788 -0.7104159
hp   -0.7761684  0.8324475  0.7909486  1.0000000 -0.44875912  0.6587479 -0.70822339 -0.7230967
drat  0.6811719 -0.6999381 -0.7102139 -0.4487591  1.00000000 -0.7124406  0.09120476  0.4402785
wt   -0.8676594  0.7824958  0.8879799  0.6587479 -0.71244065  1.0000000 -0.17471588 -0.5549157
qsec  0.4186840 -0.5912421 -0.4336979 -0.7082234  0.09120476 -0.1747159  1.00000000  0.7445354
vs    0.6640389 -0.8108118 -0.7104159 -0.7230967  0.44027846 -0.5549157  0.74453544  1.0000000
am    0.5998324 -0.5226070 -0.5912270 -0.2432043  0.71271113 -0.6924953 -0.22986086  0.1683451
gear  0.4802848 -0.4926866 -0.5555692 -0.1257043  0.69961013 -0.5832870 -0.21268223  0.2060233
carb -0.5509251  0.5269883  0.3949769  0.7498125 -0.09078980  0.4276059 -0.65624923 -0.5696071
              am       gear        carb
mpg   0.59983243  0.4802848 -0.55092507
cyl  -0.52260705 -0.4926866  0.52698829
disp -0.59122704 -0.5555692  0.39497686
hp   -0.24320426 -0.1257043  0.74981247
drat  0.71271113  0.6996101 -0.09078980
wt   -0.69249526 -0.5832870  0.42760594
qsec -0.22986086 -0.2126822 -0.65624923
vs    0.16834512  0.2060233 -0.56960714
am    1.00000000  0.7940588  0.05753435
gear  0.79405876  1.0000000  0.27407284
carb  0.05753435  0.2740728  1.00000000

b) Build Model

#Full Model
mr_model <- lm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb, data1)
summary(mr_model)

Call:
lm(formula = mpg ~ cyl + disp + hp + drat + wt + qsec + vs + 
    am + gear + carb, data = data1)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4506 -1.6044 -0.1196  1.2193  4.6271 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 12.30337   18.71788   0.657   0.5181  
cyl         -0.11144    1.04502  -0.107   0.9161  
disp         0.01334    0.01786   0.747   0.4635  
hp          -0.02148    0.02177  -0.987   0.3350  
drat         0.78711    1.63537   0.481   0.6353  
wt          -3.71530    1.89441  -1.961   0.0633 .
qsec         0.82104    0.73084   1.123   0.2739  
vs           0.31776    2.10451   0.151   0.8814  
am           2.52023    2.05665   1.225   0.2340  
gear         0.65541    1.49326   0.439   0.6652  
carb        -0.19942    0.82875  -0.241   0.8122  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.65 on 21 degrees of freedom
Multiple R-squared:  0.869, Adjusted R-squared:  0.8066 
F-statistic: 13.93 on 10 and 21 DF,  p-value: 3.793e-07
# Conclusion From the summary
# 1. Based on the t test, none of them are significant, which is potentially meaning that The Full Model is a bad choice for predicting mpg.
vif(mr_model)
      cyl      disp        hp      drat        wt      qsec        vs        am      gear 
15.373833 21.620241  9.832037  3.374620 15.164887  7.527958  4.965873  4.648487  5.357452 
     carb 
 7.908747 
# 2. Some of them has multicollinearity issures
plot(mr_model)

c)Use Added variable plot to visualize it

ols_plot_added_variable(mr_model)
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'
`geom_smooth()` using formula 'y ~ x'

d) ols regression Output

(ols_regress(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb, data1))
                        Model Summary                          
--------------------------------------------------------------
R                       0.932       RMSE                2.650 
R-Squared               0.869       Coef. Var          13.191 
Adj. R-Squared          0.807       MSE                 7.024 
Pred R-Squared          0.654       MAE                 1.723 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     978.553        10         97.855    13.932    0.0000 
Residual       147.494        21          7.024                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                   Parameter Estimates                                    
-----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig       lower     upper 
-----------------------------------------------------------------------------------------
(Intercept)    12.303        18.718                  0.657    0.518    -26.623    51.229 
        cyl    -0.111         1.045       -0.033    -0.107    0.916     -2.285     2.062 
       disp     0.013         0.018        0.274     0.747    0.463     -0.024     0.050 
         hp    -0.021         0.022       -0.244    -0.987    0.335     -0.067     0.024 
       drat     0.787         1.635        0.070     0.481    0.635     -2.614     4.188 
         wt    -3.715         1.894       -0.603    -1.961    0.063     -7.655     0.224 
       qsec     0.821         0.731        0.243     1.123    0.274     -0.699     2.341 
         vs     0.318         2.105        0.027     0.151    0.881     -4.059     4.694 
         am     2.520         2.057        0.209     1.225    0.234     -1.757     6.797 
       gear     0.655         1.493        0.080     0.439    0.665     -2.450     3.761 
       carb    -0.199         0.829       -0.053    -0.241    0.812     -1.923     1.524 
-----------------------------------------------------------------------------------------

d) Model Diagnostics Plot

Conclusion: 1. The QQ Plots show that the Full Model residuals are Normally Distributed. 2. There are 4 out of 32 outliers based on cooks D, which is kind of high proportion.

#Generate QQ Plot
qqnorm(residuals(mr_model),ylab="Residuals",main="Q-Q plot")
qqline(residuals(mr_model))


#Cook's D
ols_plot_cooksd_bar(mr_model)

ols_plot_cooksd_chart(mr_model)


#dfbetas panel
ols_plot_dfbetas(mr_model)


#dffits plot
ols_plot_dffits(mr_model)


#Studentized residuals
ols_plot_resid_stud(mr_model)


#Standardized residuals
ols_plot_resid_stand(mr_model)


#Studentized Residuals vs Leverage Plot
ols_plot_resid_lev(mr_model)


#Deleted Studentized Residual vs Fitted Values Plot
ols_plot_resid_stud_fit(mr_model)


#Hadi Plot
ols_plot_hadi(mr_model)


#Potential Residual Plot
ols_plot_resid_pot(mr_model)



2. Backward Elimination

Conclusion: Backward Elimination gives that the model mpg ~ disp + hp + wt + qsec + am is the best subset model.

backward.reg <- ols_step_backward_p(mr_model,details=TRUE)
Backward Elimination Method 
---------------------------

Candidate Terms: 

1 . cyl 
2 . disp 
3 . hp 
4 . drat 
5 . wt 
6 . qsec 
7 . vs 
8 . am 
9 . gear 
10 . carb 

We are eliminating variables based on p value...

x cyl 

Backward Elimination: Step 1 

 Variable cyl Removed 

                        Model Summary                          
--------------------------------------------------------------
R                       0.932       RMSE                2.590 
R-Squared               0.869       Coef. Var          12.891 
Adj. R-Squared          0.815       MSE                 6.708 
Pred R-Squared          0.704       MAE                 1.720 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     978.473         9        108.719    16.208    0.0000 
Residual       147.574        22          6.708                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                   Parameter Estimates                                    
-----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig       lower     upper 
-----------------------------------------------------------------------------------------
(Intercept)    10.960        13.530                  0.810    0.427    -17.100    39.020 
       disp     0.013         0.017        0.264     0.763    0.454     -0.022     0.048 
         hp    -0.022         0.021       -0.249    -1.048    0.306     -0.065     0.021 
       drat     0.835         1.536        0.074     0.544    0.592     -2.351     4.021 
         wt    -3.693         1.840       -0.599    -2.007    0.057     -7.507     0.122 
       qsec     0.842         0.687        0.250     1.227    0.233     -0.582     2.267 
         vs     0.390         1.948        0.033     0.200    0.843     -3.650     4.430 
         am     2.577         1.940        0.213     1.328    0.198     -1.447     6.601 
       gear     0.712         1.366        0.087     0.521    0.608     -2.121     3.544 
       carb    -0.220         0.789       -0.059    -0.278    0.783     -1.855     1.416 
-----------------------------------------------------------------------------------------


x vs 

Backward Elimination: Step 2 

 Variable vs Removed 

                        Model Summary                          
--------------------------------------------------------------
R                       0.932       RMSE                2.535 
R-Squared               0.869       Coef. Var          12.620 
Adj. R-Squared          0.823       MSE                 6.428 
Pred R-Squared          0.732       MAE                 1.741 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     978.204         8        122.276    19.022    0.0000 
Residual       147.843        23          6.428                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                   Parameter Estimates                                    
-----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig       lower     upper 
-----------------------------------------------------------------------------------------
(Intercept)     9.768        11.892                  0.821    0.420    -14.833    34.369 
       disp     0.012         0.016        0.250     0.753    0.459     -0.021     0.045 
         hp    -0.021         0.020       -0.238    -1.051    0.304     -0.062     0.020 
       drat     0.875         1.491        0.078     0.587    0.563     -2.210     3.960 
         wt    -3.712         1.798       -0.603    -2.064    0.050     -7.432     0.009 
       qsec     0.911         0.583        0.270     1.562    0.132     -0.295     2.117 
         am     2.524         1.881        0.209     1.342    0.193     -1.368     6.416 
       gear     0.760         1.316        0.093     0.577    0.569     -1.962     3.482 
       carb    -0.248         0.759       -0.066    -0.327    0.747     -1.819     1.323 
-----------------------------------------------------------------------------------------


x carb 

Backward Elimination: Step 3 

 Variable carb Removed 

                        Model Summary                          
--------------------------------------------------------------
R                       0.932       RMSE                2.488 
R-Squared               0.868       Coef. Var          12.382 
Adj. R-Squared          0.830       MSE                 6.189 
Pred R-Squared          0.762       MAE                 1.743 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     977.519         7        139.646    22.565    0.0000 
Residual       148.528        24          6.189                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                   Parameter Estimates                                    
-----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig       lower     upper 
-----------------------------------------------------------------------------------------
(Intercept)     9.198        11.542                  0.797    0.433    -14.624    33.020 
       disp     0.016         0.012        0.319     1.278    0.213     -0.010     0.041 
         hp    -0.025         0.016       -0.281    -1.548    0.135     -0.058     0.008 
       drat     0.810         1.450        0.072     0.559    0.582     -2.183     3.803 
         wt    -4.131         1.236       -0.671    -3.342    0.003     -6.681    -1.580 
       qsec     1.010         0.489        0.299     2.066    0.050      0.001     2.019 
         am     2.590         1.835        0.214     1.411    0.171     -1.198     6.378 
       gear     0.606         1.206        0.074     0.503    0.620     -1.883     3.095 
-----------------------------------------------------------------------------------------


x gear 

Backward Elimination: Step 4 

 Variable gear Removed 

                        Model Summary                          
--------------------------------------------------------------
R                       0.931       RMSE                2.450 
R-Squared               0.867       Coef. Var          12.196 
Adj. R-Squared          0.835       MSE                 6.004 
Pred R-Squared          0.785       MAE                 1.769 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     975.954         6        162.659    27.093    0.0000 
Residual       150.093        25          6.004                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                   Parameter Estimates                                    
-----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig       lower     upper 
-----------------------------------------------------------------------------------------
(Intercept)    10.711        10.975                  0.976    0.338    -11.894    33.315 
       disp     0.013         0.011        0.269     1.193    0.244     -0.010     0.036 
         hp    -0.022         0.015       -0.248    -1.488    0.149     -0.052     0.008 
       drat     1.021         1.367        0.091     0.746    0.462     -1.796     3.837 
         wt    -4.045         1.206       -0.657    -3.355    0.003     -6.527    -1.562 
       qsec     0.991         0.480        0.294     2.064    0.050      0.002     1.979 
         am     2.985         1.634        0.247     1.827    0.080     -0.380     6.350 
-----------------------------------------------------------------------------------------


x drat 

Backward Elimination: Step 5 

 Variable drat Removed 

                        Model Summary                          
--------------------------------------------------------------
R                       0.929       RMSE                2.429 
R-Squared               0.864       Coef. Var          12.092 
Adj. R-Squared          0.838       MSE                 5.901 
Pred R-Squared          0.798       MAE                 1.815 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     972.609         5        194.522    32.962    0.0000 
Residual       153.438        26          5.901                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    14.362         9.741                  1.474    0.152    -5.661    34.384 
       disp     0.011         0.011        0.231     1.060    0.299    -0.011     0.033 
         hp    -0.021         0.015       -0.241    -1.460    0.156    -0.051     0.009 
         wt    -4.084         1.194       -0.663    -3.420    0.002    -6.539    -1.630 
       qsec     1.007         0.475        0.299     2.118    0.044     0.030     1.984 
         am     3.470         1.486        0.287     2.336    0.027     0.416     6.525 
----------------------------------------------------------------------------------------



No more variables satisfy the condition of p value = 0.3


Variables Removed: 

x cyl 
x vs 
x carb 
x gear 
x drat 


Final Model Output 
------------------

                        Model Summary                          
--------------------------------------------------------------
R                       0.929       RMSE                2.429 
R-Squared               0.864       Coef. Var          12.092 
Adj. R-Squared          0.838       MSE                 5.901 
Pred R-Squared          0.798       MAE                 1.815 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     972.609         5        194.522    32.962    0.0000 
Residual       153.438        26          5.901                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    14.362         9.741                  1.474    0.152    -5.661    34.384 
       disp     0.011         0.011        0.231     1.060    0.299    -0.011     0.033 
         hp    -0.021         0.015       -0.241    -1.460    0.156    -0.051     0.009 
         wt    -4.084         1.194       -0.663    -3.420    0.002    -6.539    -1.630 
       qsec     1.007         0.475        0.299     2.118    0.044     0.030     1.984 
         am     3.470         1.486        0.287     2.336    0.027     0.416     6.525 
----------------------------------------------------------------------------------------



3. Forward Elimination

Conclusion: Backward Elimination gives that the model mpg ~ wt + cyl + hp is the best subset model.

forward.reg <- ols_step_forward_p(mr_model,details=TRUE)
Forward Selection Method    
---------------------------

Candidate Terms: 

1. cyl 
2. disp 
3. hp 
4. drat 
5. wt 
6. qsec 
7. vs 
8. am 
9. gear 
10. carb 

We are selecting variables based on p value...


Forward Selection: Step 1 

+ wt 

                        Model Summary                          
--------------------------------------------------------------
R                       0.868       RMSE                3.046 
R-Squared               0.753       Coef. Var          15.161 
Adj. R-Squared          0.745       MSE                 9.277 
Pred R-Squared          0.709       MAE                 2.341 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     847.725         1        847.725    91.375    0.0000 
Residual       278.322        30          9.277                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    37.285         1.878                 19.858    0.000    33.450    41.120 
         wt    -5.344         0.559       -0.868    -9.559    0.000    -6.486    -4.203 
----------------------------------------------------------------------------------------



Forward Selection: Step 2 

+ cyl 

                        Model Summary                          
--------------------------------------------------------------
R                       0.911       RMSE                2.568 
R-Squared               0.830       Coef. Var          12.780 
Adj. R-Squared          0.819       MSE                 6.592 
Pred R-Squared          0.790       MAE                 1.921 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     934.875         2        467.438    70.908    0.0000 
Residual       191.172        29          6.592                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    39.686         1.715                 23.141    0.000    36.179    43.194 
         wt    -3.191         0.757       -0.518    -4.216    0.000    -4.739    -1.643 
        cyl    -1.508         0.415       -0.447    -3.636    0.001    -2.356    -0.660 
----------------------------------------------------------------------------------------



Forward Selection: Step 3 

+ hp 

                        Model Summary                          
--------------------------------------------------------------
R                       0.918       RMSE                2.512 
R-Squared               0.843       Coef. Var          12.501 
Adj. R-Squared          0.826       MSE                 6.308 
Pred R-Squared          0.796       MAE                 1.845 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     949.427         3        316.476    50.171    0.0000 
Residual       176.621        28          6.308                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    38.752         1.787                 21.687    0.000    35.092    42.412 
         wt    -3.167         0.741       -0.514    -4.276    0.000    -4.684    -1.650 
        cyl    -0.942         0.551       -0.279    -1.709    0.098    -2.070     0.187 
         hp    -0.018         0.012       -0.205    -1.519    0.140    -0.042     0.006 
----------------------------------------------------------------------------------------



No more variables to be added.

Variables Entered: 

+ wt 
+ cyl 
+ hp 


Final Model Output 
------------------

                        Model Summary                          
--------------------------------------------------------------
R                       0.918       RMSE                2.512 
R-Squared               0.843       Coef. Var          12.501 
Adj. R-Squared          0.826       MSE                 6.308 
Pred R-Squared          0.796       MAE                 1.845 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     949.427         3        316.476    50.171    0.0000 
Residual       176.621        28          6.308                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    38.752         1.787                 21.687    0.000    35.092    42.412 
         wt    -3.167         0.741       -0.514    -4.276    0.000    -4.684    -1.650 
        cyl    -0.942         0.551       -0.279    -1.709    0.098    -2.070     0.187 
         hp    -0.018         0.012       -0.205    -1.519    0.140    -0.042     0.006 
----------------------------------------------------------------------------------------



## 4. Forward Elimination Conclusion: Backward Elimination gives that the model mpg ~ wt + cyl + hp is the best subset model.

forward.reg <- ols_step_both_p(mr_model,details=TRUE)
Stepwise Selection Method   
---------------------------

Candidate Terms: 

1. cyl 
2. disp 
3. hp 
4. drat 
5. wt 
6. qsec 
7. vs 
8. am 
9. gear 
10. carb 

We are selecting variables based on p value...


Stepwise Selection: Step 1 

+ wt 

                        Model Summary                          
--------------------------------------------------------------
R                       0.868       RMSE                3.046 
R-Squared               0.753       Coef. Var          15.161 
Adj. R-Squared          0.745       MSE                 9.277 
Pred R-Squared          0.709       MAE                 2.341 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     847.725         1        847.725    91.375    0.0000 
Residual       278.322        30          9.277                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    37.285         1.878                 19.858    0.000    33.450    41.120 
         wt    -5.344         0.559       -0.868    -9.559    0.000    -6.486    -4.203 
----------------------------------------------------------------------------------------



Stepwise Selection: Step 2 

+ cyl 

                        Model Summary                          
--------------------------------------------------------------
R                       0.911       RMSE                2.568 
R-Squared               0.830       Coef. Var          12.780 
Adj. R-Squared          0.819       MSE                 6.592 
Pred R-Squared          0.790       MAE                 1.921 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     934.875         2        467.438    70.908    0.0000 
Residual       191.172        29          6.592                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    39.686         1.715                 23.141    0.000    36.179    43.194 
         wt    -3.191         0.757       -0.518    -4.216    0.000    -4.739    -1.643 
        cyl    -1.508         0.415       -0.447    -3.636    0.001    -2.356    -0.660 
----------------------------------------------------------------------------------------

                        Model Summary                          
--------------------------------------------------------------
R                       0.911       RMSE                2.568 
R-Squared               0.830       Coef. Var          12.780 
Adj. R-Squared          0.819       MSE                 6.592 
Pred R-Squared          0.790       MAE                 1.921 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     934.875         2        467.438    70.908    0.0000 
Residual       191.172        29          6.592                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    39.686         1.715                 23.141    0.000    36.179    43.194 
         wt    -3.191         0.757       -0.518    -4.216    0.000    -4.739    -1.643 
        cyl    -1.508         0.415       -0.447    -3.636    0.001    -2.356    -0.660 
----------------------------------------------------------------------------------------



No more variables to be added/removed.


Final Model Output 
------------------

                        Model Summary                          
--------------------------------------------------------------
R                       0.911       RMSE                2.568 
R-Squared               0.830       Coef. Var          12.780 
Adj. R-Squared          0.819       MSE                 6.592 
Pred R-Squared          0.790       MAE                 1.921 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                               ANOVA                                 
--------------------------------------------------------------------
                Sum of                                              
               Squares        DF    Mean Square      F         Sig. 
--------------------------------------------------------------------
Regression     934.875         2        467.438    70.908    0.0000 
Residual       191.172        29          6.592                     
Total         1126.047        31                                    
--------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    39.686         1.715                 23.141    0.000    36.179    43.194 
         wt    -3.191         0.757       -0.518    -4.216    0.000    -4.739    -1.643 
        cyl    -1.508         0.415       -0.447    -3.636    0.001    -2.356    -0.660 
----------------------------------------------------------------------------------------

3 model

a) Forward

f_model <- lm(mpg ~ wt + cyl + hp, data1)
summary(f_model)

Call:
lm(formula = mpg ~ wt + cyl + hp, data = data1)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.9290 -1.5598 -0.5311  1.1850  5.8986 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 38.75179    1.78686  21.687  < 2e-16 ***
wt          -3.16697    0.74058  -4.276 0.000199 ***
cyl         -0.94162    0.55092  -1.709 0.098480 .  
hp          -0.01804    0.01188  -1.519 0.140015    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.512 on 28 degrees of freedom
Multiple R-squared:  0.8431,    Adjusted R-squared:  0.8263 
F-statistic: 50.17 on 3 and 28 DF,  p-value: 2.184e-11

b) Backward

#backward model
b_model <- lm(mpg ~ disp + hp + wt + qsec + am, data1)
summary(b_model)

c) Bidirectional

Conclusion: Stepwise Elimination gives that the model mpg ~ wt + cyl is the best subset model.

#backward model
bi_model <- lm(mpg ~ wt + cyl, data1)
summary(bi_model)

Call:
lm(formula = mpg ~ wt + cyl, data = data1)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.2893 -1.5512 -0.4684  1.5743  6.1004 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  39.6863     1.7150  23.141  < 2e-16 ***
wt           -3.1910     0.7569  -4.216 0.000222 ***
cyl          -1.5078     0.4147  -3.636 0.001064 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.568 on 29 degrees of freedom
Multiple R-squared:  0.8302,    Adjusted R-squared:  0.8185 
F-statistic: 70.91 on 2 and 29 DF,  p-value: 6.809e-12



5. All possible & Best Subset Regression

a) All Possible

all_p <- ols_step_all_possible(mr_model)
all_p
plot(all_p)
Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.

b) Best Subset

b.subset <- ols_step_best_subset(mr_model)
b.subset
                Best Subsets Regression                
-------------------------------------------------------
Model Index    Predictors
-------------------------------------------------------
     1         wt                                       
     2         cyl wt                                   
     3         wt qsec am                               
     4         hp wt qsec am                            
     5         disp hp wt qsec am                       
     6         disp hp drat wt qsec am                  
     7         disp hp drat wt qsec am gear             
     8         disp hp drat wt qsec am gear carb        
     9         disp hp drat wt qsec vs am gear carb     
    10         cyl disp hp drat wt qsec vs am gear carb 
-------------------------------------------------------

                                                   Subsets Regression Summary                                                    
---------------------------------------------------------------------------------------------------------------------------------
                       Adj.        Pred                                                                                           
Model    R-Square    R-Square    R-Square     C(p)        AIC        SBIC        SBC         MSEP       FPE       HSP       APC  
---------------------------------------------------------------------------------------------------------------------------------
  1        0.7528      0.7446      0.7087    11.6270    166.0294    74.3734    170.4266    296.9167    9.8572    0.3199    0.2801 
  2        0.8302      0.8185      0.7904     1.2187    156.0101    66.1903    161.8730    211.2280    7.2101    0.2354    0.2049 
  3        0.8497      0.8336      0.7946     0.1026    154.1194    65.7138    161.4481    193.9735    6.8017    0.2239    0.1933 
  4        0.8579      0.8368      0.8021     0.7900    154.3274    67.2299    163.1218    190.4637    6.8547    0.2280    0.1948 
  5        0.8637      0.8375      0.7984     1.8462    154.9740    69.3073    165.2341    189.8793    7.0080    0.2361    0.1992 
  6        0.8667      0.8347      0.7855     3.3700    156.2687    71.9258    167.9946    193.4796    7.3170    0.2502    0.2079 
  7        0.8681      0.8296      0.7619     5.1472    157.9333    74.8058    171.1250    199.7867    7.7358    0.2691    0.2198 
  8        0.8687      0.8230      0.7316     7.0496    159.7853    77.7959    174.4427    207.9040    8.2358    0.2922    0.2340 
  9        0.8689      0.8153      0.7035     9.0114    161.7271    80.8277    177.8502    217.4086    8.8041    0.3194    0.2502 
 10        0.8690      0.8066      0.6538    11.0000    163.7098    83.8728    181.2986    228.1554    9.4379    0.3512    0.2682 
---------------------------------------------------------------------------------------------------------------------------------
AIC: Akaike Information Criteria 
 SBIC: Sawa's Bayesian Information Criteria 
 SBC: Schwarz Bayesian Criteria 
 MSEP: Estimated error of prediction, assuming multivariate normality 
 FPE: Final Prediction Error 
 HSP: Hocking's Sp 
 APC: Amemiya Prediction Criteria 
plot(b.subset)

c) Analysis on All possible Regression

Conclusion: The Model Selection Criteria based on All Possible Regression gives that mpg~wt+qsec+am is the best model for mtcars

all_p$predictors[all_p$rsquare == max(all_p$rsquare)]
[1] "cyl disp hp drat wt qsec vs am gear carb"
all_p$predictors[all_p$adjr == max(all_p$adjr)]
[1] "disp hp wt qsec am"
all_p$predictors[all_p$cp == min(all_p$cp)]
[1] "wt qsec am"
all_p$predictors[all_p$aic == min(all_p$aic)]
[1] "wt qsec am"
all_p$predictors[all_p$sbic == min(all_p$sbic)]
[1] "wt qsec am"
all_p$predictors[all_p$sbc == min(all_p$sbc)]
[1] "wt qsec am"

c) Analysis on Best Subset Regression

Conclusion: The Model Selection Criteria based on All Best Subset Regression gives that mpg~wt+qsec+am is the best model for mtcars

b.subset$predictors[b.subset$rsquare == max(b.subset$rsquare)]
[1] "cyl disp hp drat wt qsec vs am gear carb"
b.subset$predictors[b.subset$adjr == max(b.subset$adjr)]
[1] "disp hp wt qsec am"
b.subset$predictors[b.subset$cp == min(b.subset$cp)]
[1] "wt qsec am"
b.subset$predictors[b.subset$aic == min(b.subset$aic)]
[1] "wt qsec am"
b.subset$predictors[b.subset$sbic == min(b.subset$sbic)]
[1] "wt qsec am"
b.subset$predictors[b.subset$sbc == min(b.subset$sbc)]
[1] "wt qsec am"



6. Cross validation

Current we have the options: 1. wt + qsec + am 2. wt + cyl 3. disp + hp + wt + qsec + am 4. wt + cyl + hp

a) Create trainControl

train.control1 <- trainControl(method = "cv", number = 5)
train.control2 <- trainControl(method = "repeatedcv", number = 5, repeats = 100)

b) One time Cross Validation

Conclusion: It is still not explicit to say which model should be chose.

#How about n-1 fold?
option1 <- train(mpg ~ wt + qsec + am, data = data1, method = "lm", trControl = train.control1)
option2 <- train(mpg ~ wt + cyl, data = data1, method = "lm", trControl = train.control1)
option3 <- train(mpg ~ disp + hp + wt + qsec + am, data = data1, method = "lm", trControl = train.control1)
option4 <- train(mpg ~ wt + cyl + hp, data = data1, method = "lm", trControl = train.control1)
print(option1)
Linear Regression 

32 samples
 3 predictor

No pre-processing
Resampling: Cross-Validated (5 fold) 
Summary of sample sizes: 27, 25, 26, 25, 25 
Resampling results:

  RMSE      Rsquared   MAE     
  2.369457  0.8611764  1.982344

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option2)
Linear Regression 

32 samples
 2 predictor

No pre-processing
Resampling: Cross-Validated (5 fold) 
Summary of sample sizes: 25, 25, 25, 27, 26 
Resampling results:

  RMSE      Rsquared   MAE     
  2.467026  0.8398594  2.007533

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option3)
Linear Regression 

32 samples
 5 predictor

No pre-processing
Resampling: Cross-Validated (5 fold) 
Summary of sample sizes: 24, 26, 26, 25, 27 
Resampling results:

  RMSE      Rsquared   MAE     
  2.665735  0.8697989  2.170045

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option4)
Linear Regression 

32 samples
 3 predictor

No pre-processing
Resampling: Cross-Validated (5 fold) 
Summary of sample sizes: 25, 26, 26, 26, 25 
Resampling results:

  RMSE      Rsquared   MAE     
  2.419125  0.8294988  2.082439

Tuning parameter 'intercept' was held constant at a value of TRUE
tot <- 1
count <- data.frame(rmse_count = c(0,0,0,0), rsq_count = c(0,0,0,0), mae_count = c(0,0,0,0), total_num = c(tot,tot,tot,tot))
rmse_step <- data.frame(ind = c(1,2,3,4), rmse = c(option1$results$RMSE,option2$results$RMSE,option3$results$RMSE,option4$results$RMSE))
ind_max1 <- rmse_step$ind[rmse_step$rmse == min(rmse_step$rmse)]
count$rmse_count[ind_max1] <- count$rmse_count[ind_max1] + 1
  
rsq_step <- data.frame(ind = c(1,2,3,4), rsq = c(option1$results$Rsquared,option2$results$Rsquared,option3$results$Rsquared,option4$results$Rsquared))
ind_max2 <- rsq_step$ind[rsq_step$rsq == max(rsq_step$rsq)]
count$rsq_count[ind_max2] <- count$rsq_count[ind_max2] + 1
  
mae_step <- data.frame(ind = c(1,2,3,4), mae = c(option1$results$MAE,option2$results$MAE,option3$results$MAE,option4$results$MAE))
ind_max3 <- mae_step$ind[mae_step$mae == min(mae_step$mae)]
count$mae_count[ind_max3] <- count$mae_count[ind_max3] + 1
count

c) Repeated Cross Validation

Conclusion: We have to choose 4th Model.

#How about n-1 fold?
option1 <- train(mpg ~ wt + qsec + am, data = data1, method = "lm", trControl = train.control2)
option2 <- train(mpg ~ wt + cyl, data = data1, method = "lm", trControl = train.control2)
option3 <- train(mpg ~ disp + hp + wt + qsec + am, data = data1, method = "lm", trControl = train.control2)
option4 <- train(mpg ~ wt + cyl + hp, data = data1, method = "lm", trControl = train.control2)
print(option1)
Linear Regression 

32 samples
 3 predictor

No pre-processing
Resampling: Cross-Validated (5 fold, repeated 100 times) 
Summary of sample sizes: 27, 25, 25, 25, 26, 26, ... 
Resampling results:

  RMSE      Rsquared   MAE     
  2.616779  0.8461208  2.226053

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option2)
Linear Regression 

32 samples
 2 predictor

No pre-processing
Resampling: Cross-Validated (5 fold, repeated 100 times) 
Summary of sample sizes: 27, 24, 25, 26, 26, 26, ... 
Resampling results:

  RMSE      Rsquared   MAE     
  2.574947  0.8618947  2.099273

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option3)
Linear Regression 

32 samples
 5 predictor

No pre-processing
Resampling: Cross-Validated (5 fold, repeated 100 times) 
Summary of sample sizes: 27, 25, 27, 24, 25, 25, ... 
Resampling results:

  RMSE      Rsquared   MAE     
  2.600129  0.8491927  2.222816

Tuning parameter 'intercept' was held constant at a value of TRUE
print(option4)
Linear Regression 

32 samples
 3 predictor

No pre-processing
Resampling: Cross-Validated (5 fold, repeated 100 times) 
Summary of sample sizes: 27, 25, 24, 26, 26, 27, ... 
Resampling results:

  RMSE      Rsquared   MAE     
  2.553296  0.8657582  2.081244

Tuning parameter 'intercept' was held constant at a value of TRUE
tot <- 1
count <- data.frame(rmse_count = c(0,0,0,0), rsq_count = c(0,0,0,0), mae_count = c(0,0,0,0), total_num = c(tot,tot,tot,tot))
rmse_step <- data.frame(ind = c(1,2,3,4), rmse = c(option1$results$RMSE,option2$results$RMSE,option3$results$RMSE,option4$results$RMSE))
ind_max1 <- rmse_step$ind[rmse_step$rmse == min(rmse_step$rmse)]
count$rmse_count[ind_max1] <- count$rmse_count[ind_max1] + 1
  
rsq_step <- data.frame(ind = c(1,2,3,4), rsq = c(option1$results$Rsquared,option2$results$Rsquared,option3$results$Rsquared,option4$results$Rsquared))
ind_max2 <- rsq_step$ind[rsq_step$rsq == max(rsq_step$rsq)]
count$rsq_count[ind_max2] <- count$rsq_count[ind_max2] + 1
  
mae_step <- data.frame(ind = c(1,2,3,4), mae = c(option1$results$MAE,option2$results$MAE,option3$results$MAE,option4$results$MAE))
ind_max3 <- mae_step$ind[mae_step$mae == min(mae_step$mae)]
count$mae_count[ind_max3] <- count$mae_count[ind_max3] + 1
count





---
title: "MTCRT DataSet Analysis"
date: "12/06"
author: "[Chenjia Li]"
output: html_notebook
---

## Import the dataset and other packages
```{r}
#rm(list = ls())

#packages
library(bayesrules)
library(randomForest)
library(rpart)
library(tree)
library(pROC)
library(mgcv)
library(ISLR)
library(dplyr)
library(tidyverse)
library(faraway)
library(olsrr)
library(caret)


#Data Set
data1 <- mtcars
summary(data1)
head(data1)
```



<br>
<br>


## 1. OLS

#### a) Visualize the data using Pairs to see if there is any collinearity
```{r}
pairs(data1)

# Conclusion
# Some of the predictors are linearly depedent based on the graph. (ex. mpg~drat, hp~wt)

cor(data1)
```

#### b) Build Model
```{r}
#Full Model
mr_model <- lm(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb, data1)
summary(mr_model)

# Conclusion From the summary
# 1. Based on the t test, none of them are significant, which is potentially meaning that The Full Model is a bad choice for predicting mpg.
vif(mr_model)
# 2. Some of them has multicollinearity issures
```

```{r}
plot(mr_model)
```
#### c)Use Added variable plot to visualize it
```{r}
ols_plot_added_variable(mr_model)
```

#### d) ols regression Output
```{r}
(ols_regress(mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb, data1))
```

#### d) Model Diagnostics Plot
Conclusion: 
1. The QQ Plots show that the Full Model residuals are Normally Distributed. 
2. There are 4 out of 32 outliers based on cooks D, which is kind of high proportion.

```{r}
#Generate QQ Plot
qqnorm(residuals(mr_model),ylab="Residuals",main="Q-Q plot")
qqline(residuals(mr_model))

#Cook's D
ols_plot_cooksd_bar(mr_model)
ols_plot_cooksd_chart(mr_model)

#dfbetas panel
ols_plot_dfbetas(mr_model)

#dffits plot
ols_plot_dffits(mr_model)

#Studentized residuals
ols_plot_resid_stud(mr_model)

#Standardized residuals
ols_plot_resid_stand(mr_model)

#Studentized Residuals vs Leverage Plot
ols_plot_resid_lev(mr_model)

#Deleted Studentized Residual vs Fitted Values Plot
ols_plot_resid_stud_fit(mr_model)

#Hadi Plot
ols_plot_hadi(mr_model)

#Potential Residual Plot
ols_plot_resid_pot(mr_model)
```

<br>
<br>

## 2. Backward Elimination
Conclusion: Backward Elimination gives that the model mpg ~ disp + hp + wt + qsec + am is the best subset model.

```{r}
backward.reg <- ols_step_backward_p(mr_model,details=TRUE)
```



<br>
<br>

## 3. Forward Elimination
Conclusion: Backward Elimination gives that the model mpg ~ wt + cyl + hp is the best subset model.
```{r}
forward.reg <- ols_step_forward_p(mr_model,details=TRUE)
```

<br>
<br>
## 4. Forward Elimination
Conclusion: Backward Elimination gives that the model mpg ~ wt + cyl + hp is the best subset model.
```{r}
forward.reg <- ols_step_both_p(mr_model,details=TRUE)
```



## 3 model 
#### a) Forward
```{r}
f_model <- lm(mpg ~ wt + cyl + hp, data1)
summary(f_model)
```

#### b) Backward
```{r}
#backward model
b_model <- lm(mpg ~ disp + hp + wt + qsec + am, data1)
summary(b_model)
```
#### c) Bidirectional
Conclusion: Stepwise Elimination gives that the model mpg ~ wt + cyl is the best subset model.
```{r}
#backward model
bi_model <- lm(mpg ~ wt + cyl, data1)
summary(bi_model)
```

<br>
<br>


## 5. All possible & Best Subset Regression
#### a) All Possible
```{r}
all_p <- ols_step_all_possible(mr_model)
all_p
plot(all_p)
```

#### b) Best Subset
```{r}
b.subset <- ols_step_best_subset(mr_model)
b.subset
plot(b.subset)
```
#### c) Analysis on All possible Regression
Conclusion: The Model Selection Criteria based on All Possible Regression gives that mpg~wt+qsec+am is the best model for mtcars
```{r}
all_p$predictors[all_p$rsquare == max(all_p$rsquare)]
all_p$predictors[all_p$adjr == max(all_p$adjr)]
all_p$predictors[all_p$cp == min(all_p$cp)]
all_p$predictors[all_p$aic == min(all_p$aic)]
all_p$predictors[all_p$sbic == min(all_p$sbic)]
all_p$predictors[all_p$sbc == min(all_p$sbc)]
```
#### c) Analysis on Best Subset Regression
Conclusion: The Model Selection Criteria based on All Best Subset Regression gives that mpg~wt+qsec+am is the best model for mtcars
```{r}
b.subset$predictors[b.subset$rsquare == max(b.subset$rsquare)]
b.subset$predictors[b.subset$adjr == max(b.subset$adjr)]
b.subset$predictors[b.subset$cp == min(b.subset$cp)]
b.subset$predictors[b.subset$aic == min(b.subset$aic)]
b.subset$predictors[b.subset$sbic == min(b.subset$sbic)]
b.subset$predictors[b.subset$sbc == min(b.subset$sbc)]
```


<br>
<br>

## 6. Cross validation
Current we have the options:
1. wt + qsec + am
2. wt + cyl
3. disp + hp + wt + qsec + am
4. wt + cyl + hp

#### a) Create trainControl
```{r}
train.control1 <- trainControl(method = "cv", number = 5)
train.control2 <- trainControl(method = "repeatedcv", number = 5, repeats = 100)
```


#### b) One time Cross Validation
Conclusion: It is still not explicit to say which model should be chose.
```{r}
#How about n-1 fold?
option1 <- train(mpg ~ wt + qsec + am, data = data1, method = "lm", trControl = train.control1)
option2 <- train(mpg ~ wt + cyl, data = data1, method = "lm", trControl = train.control1)
option3 <- train(mpg ~ disp + hp + wt + qsec + am, data = data1, method = "lm", trControl = train.control1)
option4 <- train(mpg ~ wt + cyl + hp, data = data1, method = "lm", trControl = train.control1)
print(option1)
print(option2)
print(option3)
print(option4)
```

```{r}
tot <- 1
count <- data.frame(rmse_count = c(0,0,0,0), rsq_count = c(0,0,0,0), mae_count = c(0,0,0,0), total_num = c(tot,tot,tot,tot))
rmse_step <- data.frame(ind = c(1,2,3,4), rmse = c(option1$results$RMSE,option2$results$RMSE,option3$results$RMSE,option4$results$RMSE))
ind_max1 <- rmse_step$ind[rmse_step$rmse == min(rmse_step$rmse)]
count$rmse_count[ind_max1] <- count$rmse_count[ind_max1] + 1
  
rsq_step <- data.frame(ind = c(1,2,3,4), rsq = c(option1$results$Rsquared,option2$results$Rsquared,option3$results$Rsquared,option4$results$Rsquared))
ind_max2 <- rsq_step$ind[rsq_step$rsq == max(rsq_step$rsq)]
count$rsq_count[ind_max2] <- count$rsq_count[ind_max2] + 1
  
mae_step <- data.frame(ind = c(1,2,3,4), mae = c(option1$results$MAE,option2$results$MAE,option3$results$MAE,option4$results$MAE))
ind_max3 <- mae_step$ind[mae_step$mae == min(mae_step$mae)]
count$mae_count[ind_max3] <- count$mae_count[ind_max3] + 1
count
```



#### c) Repeated Cross Validation
Conclusion: We have to choose 4th Model.
```{r}
#How about n-1 fold?
option1 <- train(mpg ~ wt + qsec + am, data = data1, method = "lm", trControl = train.control2)
option2 <- train(mpg ~ wt + cyl, data = data1, method = "lm", trControl = train.control2)
option3 <- train(mpg ~ disp + hp + wt + qsec + am, data = data1, method = "lm", trControl = train.control2)
option4 <- train(mpg ~ wt + cyl + hp, data = data1, method = "lm", trControl = train.control2)
print(option1)
print(option2)
print(option3)
print(option4)
```

```{r}
tot <- 1
count <- data.frame(rmse_count = c(0,0,0,0), rsq_count = c(0,0,0,0), mae_count = c(0,0,0,0), total_num = c(tot,tot,tot,tot))
rmse_step <- data.frame(ind = c(1,2,3,4), rmse = c(option1$results$RMSE,option2$results$RMSE,option3$results$RMSE,option4$results$RMSE))
ind_max1 <- rmse_step$ind[rmse_step$rmse == min(rmse_step$rmse)]
count$rmse_count[ind_max1] <- count$rmse_count[ind_max1] + 1
  
rsq_step <- data.frame(ind = c(1,2,3,4), rsq = c(option1$results$Rsquared,option2$results$Rsquared,option3$results$Rsquared,option4$results$Rsquared))
ind_max2 <- rsq_step$ind[rsq_step$rsq == max(rsq_step$rsq)]
count$rsq_count[ind_max2] <- count$rsq_count[ind_max2] + 1
  
mae_step <- data.frame(ind = c(1,2,3,4), mae = c(option1$results$MAE,option2$results$MAE,option3$results$MAE,option4$results$MAE))
ind_max3 <- mae_step$ind[mae_step$mae == min(mae_step$mae)]
count$mae_count[ind_max3] <- count$mae_count[ind_max3] + 1
count
```




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